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Thursday, April 26, 2012

Srinivasa Ramanujan was one of India's greatest mathematical geniuses.
He made contributions to the analytical theory of numbers and worked on
elliptic functions, continued fractions, and infinite series.
Ramanujan was born in his grandmother's house in Erode on December 22,
1887. When Ramanujan was a year old his mother took him to the town of
Kumbakonam, near Madras. His father worked in Kumbakonam as a clerk in a
cloth merchant's shop.
When he was five years old, Ramanujan went to the primary school in Kumbakonam although he would attend several different primary schools
before entering the Town High School in Kumbakonam in January 1898. At
the Town High School, Ramanujan did well in all his school subjects and
showed himself as a talented student. In 1900 he began to work on his
own on mathematics summing geometric and arithmetic series.
Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic.
It was in the Town High School that Ramanujan came across a mathematics
book by G. S. Carr called Synopsis of Elementary Results in Pure
Mathematics. Ramanujan used this to teach himself mathematics. The book
contained theorems, formulas and short proofs. It also contained an
index to papers on pure mathematics.
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the numbers, which is entirely his own independent discovery.
Ramanujan, on the strength of his good schoolwork, was given a
scholarship to the Government College in Kumbakonam, which he entered in
1904. However the following year his scholarship was not renewed
because Ramanujan devoted more and more of his time to mathematics and
neglected his other subjects. Without money he was soon in difficulties
and, without telling his parents, he ran away to the town of
Vizagapatnam. He continued his mathematical work, and at this time he
worked on hyper geometric series and investigated relations between
integrals and series. He learned later that he had been studying
elliptic functions.
In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College.
His wanted to pass the First Arts examination that would allow him to
be admitted to the University of Madras. He attended lectures at
Pachaiyappa's College but became ill after three months study. He took
the First Arts examination after having left the course. He passed in
mathematics but failed all his other subjects and therefore failed the
examination. This meant that he could not enter the University of
Madras. In the following years he worked on mathematics developing his
own ideas without any help and without any real idea of the then current
research topics other than that provided by Carr's book.
Continuing his mathematical work Ramanujan studied continued fractions
and divergent series in 1908. At this stage he became seriously ill and
underwent an operation in April 1909 after which he took him some
considerable time to recover. He married on 14 July 1909 when his mother
arranged for him to marry a ten year old girl Janaki Ammal. Ramanujan
did not live with his wife until she was twelve years old.
Ramanujan continued to develop his mathematical ideas and began to pose
problems and solve problems in the Journal of the Indian Mathematical
Society. He developed relations between elliptic modular equations in
1910. After publication of a brilliant research paper on Bernoulli
numbers in 1911 in the Journal of the Indian Mathematical Society he
gained recognition for his work. Even though he lacked a university
education, he was becoming well known in the Madras area as a
mathematical genius.
In 1911, Ramanujan approached the founder of the Indian Mathematical
Society for advice on a job. After this he was appointed to his first
job, a temporary post in the Accountant General's Office in Madras. He
asked Ramachandra Rao, the founder member of the Indian Mathematical
Society who had helped start the mathematics library, about a job in
mathematics.
Ramachandra Rao told him to return to Madras and he tried,
unsuccessfully, to arrange a scholarship for Ramanujan. In 1912
Ramanujan applied for the post of clerk in the accounts section of the
Madras Port Trust.

Ramanujan was appointed to the post of clerk and began his duties on 1
March 1912. Ramanujan was quite lucky to have a number of people working
round him with training in mathematics. In fact the Chief Accountant
for the Madras Port Trust, S N Aiyar, was trained as a mathematician and
published a paper On the distribution of primes in 1913 on Ramanujan's
work. The professor of civil engineering at the Madras Engineering
College, T. Griffith was also interested in Ramanujan's abilities and,
having been educated at University College London, knew the professor of
mathematics there, namely M. Hill. He wrote to Hill on 12 November 1912
sending some of Ramanujan's work and a copy of his 1911 paper on
Bernoulli numbers.
Hill replied in a fairly encouraging way but showed that he had failed
to understand Ramanujan's results on divergent series. The
recommendation to Ramanujan that he read Bromwich's Theory of infinite
series did not please Ramanujan much. Ramanujan wrote to E. W. Hobson
and H. F. Baker trying to interest them in his results but neither
replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy
of his 1910 book Orders of infinity.
Hardy studied the long list of unproved theorems which Ramanujan enclosed with his letter.

The University of Madras gave Ramanujan a scholarship in May 1913 for
two years and, in 1914, Hardy brought Ramanujan to Trinity College,
Cambridge, to begin a surprising collaboration.
Ramanujan sailed from India on March 17, 1914. He arrived in London on
April 14, 1914. Right from the beginning Ramanujan had problems with his
diet, his religion prevented him from eating the foods available at
Cambridge University. The outbreak of World War I made obtaining special
items of food harder and it was not long before Ramanujan had health
problems.
Right from the start Ramanujan's collaboration with Hardy led to
important results. Hardy was, however, unsure how to approach the
problem of Ramanujan's lack of formal education.

The war soon took many away on war duty but Hardy remained in Cambridge
to work with Ramanujan. Even in his first winter in England, Ramanujan
was ill and he wrote in March 1915 that he had been ill due to the
winter weather and had not been able to publish anything for five
months. What he did publish was the work he did in England, the decision
having been made that the results he had obtained while in India, many
of which he had communicated to Hardy in his letters, would not be
published until the war had ended.
On March 16, 1916 Ramanujan graduated from Cambridge with a Bachelor of
Science by Research (the degree was called a Ph.D. in 1920). Ramanujan's
thesis was on Highly composite numbers and consisted of seven of his
papers published in England.
Ramanujan fell seriously ill in 1917 and his doctors feared that he
would die. He did improve a little by September but spent most of his
time in various nursing homes
On 18 February 1918 Ramanujan was elected a fellow of the Cambridge
Philosophical Society and then three days later, the greatest honor that
he would receive, his name appeared on the list for election as a
fellow of the Royal Society of London. He had been proposed by an
impressive list of mathematicians. His election as a fellow of the Royal
Society was confirmed on May 2, 1918, then on October 10, 1918 he was
elected a Fellow of Trinity College Cambridge, the fellowship to run for
six years.
The honors which were given to Ramanujan seemed to help his health
improve a little and he improved his efforts at producing mathematics.
By the end of November 1918 Ramanujan's health had greatly improved. .

Ramanujan sailed to India on February 27, 1919 arriving on March 13.
However his health was very poor and, despite medical treatment, he died
there the following year.
The letters Ramanujan wrote to Hardy in 1913 had contained many
fascinating results. Ramanujan worked out the Riemann series, the
elliptic integrals, hyper geometric series and functional equations of
the zeta function. Despite many brilliant results, some of his theorems
on prime numbers were completely wrong.
Ramanujan independently discovered results of Gauss, Kummer and others
on hypergeometric series. Ramanujan's own work on partial sums and
products of hypergeometric series have led to major development in the
topic. Perhaps his most famous work was on the number p(n) of partitions
of an integer n into summands. MacMahon had produced tables of the
value of p(n) for small numbers n, and Ramanujan used this numerical
data to conjecture some remarkable properties some of which he proved
using elliptic functions. Others were only proved after Ramanujan's
death.
In a joint paper with Hardy, Ramanujan gave an asymptotic formula for
p(n). It had the remarkable property that it appeared to give the
correct value of p(n), and that was later proved by Rademacher.
Ramanujan left a number of unpublished notebooks filled with theorems
that mathematicians have continued to study. G N Watson, Mason Professor
of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers
under the general title Theorems stated by Ramanujan and in all he
published nearly 30 papers, which were inspired by Ramanujan's work.
Hardy passed on to Watson the large number of manuscripts of Ramanujan
that he had, both written before 1914 and some written in Ramanujan's
last year in India before his death.